We extend the lightning Laplace solver (Gopal and Trefethen, SINUM 2019) to unbounded domains and to the biharmonic equation. Illustrating the high accuracy of such methods, we get beautiful contour plots of Moffatt eddies.

The QDWH algorithm can compute the polar decomposition of a matrix in a stable and efficient way. We generalize this method in order to compute generalized polar decompositions with respect to signature matrices. Here, the role of the QR decomposition is played by the hyperbolic QR decomposition. However, it doesn't show the same favorable properties concerning stability as its orthogonal counterpart. Remedies are found by exploiting connections to the LDL^T factorization and by employing well-conditioned permuted graph bases. The computed polar decomposition is used to formulate a structure-preserving spectral divide-and-conquer method for pseudosymmetric matrices. Applications of this method are found in computational quantum physics, where eigenvalues and eigenvectors describe optical properties of condensed matter or molecules. Additional properties guarantee fast convergence and a reduction to symmetric definite eigenvalue problems after just one step of spectral divide-and-conquer.

We show that the scalability challenges of Global Optimisation algorithms can be overcome for functions with low effective dimensionality, which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations. We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al. (2016). Using tools from random matrix theory and conic integral geometry, we investigate the efficacy and convergence of our random subspace embeddings approach, in a static and/or adaptive formulation. We illustrate our algorithmic proposals and theoretical findings numerically, using state of the art global solvers. This work is joint with Coralia Cartis.
 

Subspace correction (SSC) methods are based on a "divide and conquer" strategy, where a global problem is divided into a sequence of local ones. This framework includes iterative methods such as Jacobi iteration, domain decomposition, and multigrid methods. We are interested in nonlinear PDEs exhibiting highly localized nonlinearities, for which Newton's method can take many iterations. Instead of doing all this global work, nonlinear SSC methods tackle the localized nonlinearities within subproblems. In this presentation, we describe the SSC framework, and investigate combining Newton's method with nonlinear SSC methods to solve a regularized Stefan problem.
 

Vandermonde matrices are exponentially ill-conditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem.

I will discuss the origin of the choice of global structure
--- or equivalently, the choice for which higher p-form symmetries are
present in the theory --- for various (Lagrangian and non-Lagrangian)
field theories in terms of their realization in IIB and M-theory. I
will explain how this choice on the field theory side can be traced
back to the fact that fluxes in string/M-theory do not commute in the
presence of torsion. I will illustrate how these ideas provide a
stringy explanation for the fact that six-dimensional (2,0) and (1,0)
theories generically have a partition vector (as opposed to a partition
function) and explain how this reproduces the classification of N=4
theories provided by Aharony, Seiberg and Tachikawa. Time permitting, I
will also explain how to use these ideas to obtain the algebra of
higher p-form symmetries for 5d SCFTs arising from M-theory at
arbitrary isolated toric singularities, and to classify global forms
for various 4d theories in the presence of duality defects.

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no known cure. Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. In this talk, I will show that by linking new mathematical theories to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions. In particular, I will outline interesting mathematical problems and ideas that naturally appear in the process.

Heterogeneity in Space and Time: Novel Dispersion Relations in Morphogenesis

Dr. Andrew Krause

Motivated by recent work with biologists, I will showcase some results on Turing instabilities in complex domains. This is scientifically related to understanding developmental tuning in the whiskers of mice, and in synthetic quorum-sensing patterning of bacteria. Such phenomena are typically modelled using reaction-diffusion systems of morphogens, and one is often interested in emergent spatial and spatiotemporal patterns resulting from instabilities of a homogeneous equilibrium. In comparison to the well-known effects of how advection or manifold structure impacts the modes which may become unstable in such systems, I will present results on instabilities in heterogeneous systems, reaction-diffusion systems on evolving manifolds, as well as layered reaction-diffusion systems. These contexts require novel formulations of classical dispersion relations, and may have applications beyond developmental biology, such as in understanding niche formation for populations of animals in heterogeneous environments. These approaches also help close the vast gap between the simplistic theory of instability-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in concretely demonstrating such a theory's applicability in real biological systems.
 

Cavity flow characteristics and applications to kidney stone removal

Dr. Jessica Williams

Ureteroscopy is a minimally invasive surgical procedure for the removal of kidney stones. A ureteroscope, containing a hollow, cylindrical working channel, is inserted into the patient's kidney. The renal space proximal to the scope tip is irrigated, to clear stone particles and debris, with a saline solution that flows in through the working channel. We consider the fluid dynamics of irrigation fluid within the renal pelvis, resulting from the emerging jet through the working channel and return flow through an access sheath . Representing the renal pelvis as a two-dimensional rectangular cavity, we investigate the effects of flow rate and cavity size on flow structure and subsequent clearance time of debris. Fluid flow is modelled with the steady incompressible Navier-Stokes equations, with an imposed Poiseuille profile at the inlet boundary to model the jet of saline, and zero-stress conditions on the outlets. The resulting flow patterns in the cavity contain multiple vortical structures. We demonstrate the existence of multiple solutions dependent on the Reynolds number of the flow and the aspect ratio of the cavity using complementary numerical simulations and PIV experiments. The clearance of an initial debris cloud is simulated via solutions to an advection-diffusion equation and we characterise the effects of the initial position of the debris cloud within the vortical flow and the Péclet number on clearance time. With only weak diffusion, debris that initiates within closed streamlines can become trapped. We discuss a flow manipulation strategy to extract debris from vortices and decrease washout time.

 

Nonlinear generalizations of the Schrödinger equation are of wide applicability to a range of areas including atomic and optical systems, 
plasma physics and water waves.  In this  talk we revisit some principal excitations in atomic and optical systems (such as Bose-Einstein condensates and photo-refractive crystals), namely dark solitonic fronts in single-component systems, and dark-bright waves in multi-component systems. Upon introducing them and explaining their existence and stability properties in one spatial dimension, we will extend them both in the form of stripes and in that rings in two-dimensions, presenting an alternative (adiabatic-invariant based) formulation of their stability and excitations. We will explore their filamentary dynamics, as well as the states that emerge from their transverse (snaking) instability. Then, we will consider these structures even in three dimensions, in the form of planar, as well as spherical shell wave patterns and generalize our adiabatic invariant formulation there. Finally, time permitting, we will give some glimpses of how some of these dynamical features in 1d and 2d generalize in a multi-orbital, time-dependent quantum setting.